### What is the formula for probability density?

Probability density: f(x)=(1/2√π)exp{-(x-3)²/2*2}

Based on the expression for the normal probability density function in the question you can immediately get the mathematical expectation and the variance of the random variable:

Mathematical Expectation: μ=3

Variance: σ²=2

The continuous type random variable’s The probability density function (which can be shortened to density function when it is not so confusing) is a function that describes the likelihood that the output value of this random variable, will be in the vicinity of some definite point of value.

And the probability that the value of the random variable falls within a certain region is the integral of the probability density function over that region. When a probability density function exists, the cumulative distribution function is the integral of the probability density function. The probability density function is usually labeled in lowercase.

Extended information:

The probability density function of random data represents the probability that the instantaneous amplitude falls within a specified range. It is therefore a function of the magnitude. It varies with the amplitude of the range taken.

The probability density function f(x) has the following properties:

(1)f(x)≧0;

(2)∫f(x)d(x)=1;

(3)P(a<X≦b)=∫f(x)dx.

### How to find the probability density function? How to find the probability density function?

1, first find the distribution function:

Y must be distributed on (1, e), X = ln(Y) obeys a uniform distribution

F(X)=P(x<=X)=X; //X obeys a uniform distribution on (0,1)

P(ln(y)<=X)=X; //substituting x=ln(y). Note that it is lowercase

P(y<=e^X)=X; // the internal condition transforms to take y as the variable

P(y<=Y)=ln(Y); // substituting X=ln(Y), and note that it is uppercase

i.e. F(Y)=P(y<=Y)=ln(Y).

2, then find the probability density:

f(y)=F'(Y)=1/Y; //probability density is the derivative of the distribution function

3, check the values of the Y variable

No overlap, no exceedance, the original solution is correct